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12.3: Monomials and Powers

Introduction

The Laboratory

One of the places that the students were able to visit when they went downtown was a laboratory at the city college. Downtown, the city college had some of its classrooms and one of the classrooms was a laboratory.

“This is a good friend of mine Professor Smith,” Mr. Travis said introducing the students to a woman with blonde hair and a wide smile.

You have already seen exponents, but let’s review the definitions of some words so that we can get started. First, let’s look at some of the parts of a term.

In the monomial above, the 7 is called thecoefficient, the is thevariable, and the 3 is theexponent.

We can say that the monomial has a power of 3 or is to the power.

Remember what we said about the coefficient of a variable like —if there is no visible coefficient, then the coefficient is an unwritten 1. You could write “” but it is not necessary. Similarly, if there is no exponent on a coefficient or variable, then you can think of it as having an unwritten exponent of 1. So 7 could be written as . The constant 7 then, is to the power.

Also, the exponent is applied to the constant, variable, or quantity that is directly to its left. That value is called the base. In the monomial above, the base is . The exponent, in this case, is not applied to the 7 because it is not directly to the left of the exponent.

What is theexponent? It’s a shortcut. It’s a way of writing many multiplications in a simpler way. In the monomial above, , the 3 indicates that the variable is multiplied by itself three times.

You can see the amount of space that is saved by using the exponent. When we write all of the multiplications instead of using the exponent, it is called theexpanded form. You can see that it is, indeed, expanded—it takes much more space to write. Imagine if the exponent were greater, like .

The exponent truly saves a lot of space! Let’s look at an example where we write an expression out into expanded form.

Example

Here we have written the expression out into expanded form.

Now, use the commutative property of multiplication to change the order of the factors so that similar factors are next to each other.

This may seem like a cumbersome long way to work, but it is accurate. There is a simpler way though-let’s take a look at the next section.

II. Recognize and Apply the Power of a Product Property to Numeric and Variable Monomial Expressions

In the previous section, we said that an exponent is applied to the constant, variable, or quantity that is directly to its left. However, we only applied exponents to single variables. Exponents can also be applied to products using parentheses.

Take the following example:

If we apply the exponent 4 to whatever is directly to its left, we would apply it to the parentheses, not just the . The parentheses are directly to the left of the 4. This indicates that the entire product in the parentheses is taken to the power. As in the previous section, we can write this in expanded form.

Now we multiply the monomials as we have already learned—by placing like factors next to each other, multiplying the coefficients, and simplifying using exponents.

This is thePower of a Product Propertywhich says, for any nonzero numbers and and any integer

Example

Example

You can see that whether we have positive or negative integers or both, we can still use the Power of a Product Property. You may have already noticed a pattern with the exponents and the final product. When you multiply like bases, there is another shortcut—you can add the exponents of like bases. Another way of saying it is:

Example

Write the definition of this property and one example down in your notebook.

III. Recognize and Appy the Power of a Quotient Property to Numeric and Variable Monomial Expressions

This may sound confusing, but in math, we can rewrite this as or . We can use exponents with fractions or quotients, too. In order to answer the question above, we would multiply the numerators and denominators across, like this: . Half of a half of a half of a half is one sixteenth. Once again, we have repeating multiplication of the same number which we could write more easily as .

ThePower of a Quotient Propertysays that for any nonzero numbers and and any integer :

Example

You can see in this example that we have simplified the expression by figuring out what five to the fourth is and what three to the fourth is. The next step in this problem would be to divide.

Example

This problem has different variables, so this is as far as we can take this problem.

IV. Recognize and Appy the Power of a Power Property to Numeric and Variable Monomial Expressions

We have raised monomials to a power, products to a power, and quotients to a power. You can see that exponents are a useful tool in simplifying expressions. If you follow the rules of exponents, the patterns become clear. We have already seen powers taken to a power. For example, look at the quotient:

If you focus on just the numerator, you can see that . You can get the exponent 28 by multiplying 7 and 4. This is an example of thePower of a Power Propertywhich says for any nonzero numbers and and any integer :

Example

Example

Example

Combine the Power of a Quotient Property and the Power of a Power Property.

Now let’s go back and work on the problem from the introduction.

Real-Life Example Completed

The Laboratory

Here is the original problem once again. Reread it and then solve for the number of grams in the sample.

One of the places that the students were able to visit when they went downtown was a laboratory at the city college. Downtown, the city college had some of its classrooms and one of the classrooms was a laboratory.

“This is a good friend of mine Professor Smith,” Mr. Travis said introducing the students to a woman with blonde hair and a wide smile.